Add the solenoid.
You should start as suggested, by cleaning all of the connections including the fuse. The basic problem remains, the wiring is too light and there are too many connections(at least eight). I had replaced everything: wiring, battery, battery cables, start button, key switch. And I had the starter tested. The problem remained, intermittent starting when I could least afford it. A few years ago I placed a sketch of the fix in the Photo Forum, see Related Link. Part #4731774 at West Marine(page 653) will work.